Up until recently, Riemannian geometry and basic topology were not
included, even by departments or faculties of mathematics, as compulsory
subjects in a university-level mathematical education. The standard courses
in the classical differential geometry of curves and surfaces which were
given instead (and still are given in some places) gradually came to be viewed
as anachronisms. However, there has been hitherto no unanimous agreement
as to exactly how such courses should be brought up to date, that is to say,
which parts of modern geometry should be regarded as absolutely essential
to a modern mathematical education, and what might be the appropriate
level of abstractness of their exposition.
The task of designing a modernized course in geometry was begun in 1971
in the mechanics division of the Faculty of Mechanics and Mathematics
of Moscow State University. The subject-matter and level of abstractness
of its exposition were dictated by the view that, in addition to the geometry
of curves and surfaces, the following topics are certainly useful in the various
areas of application of mathematics (especially in elasticity and relativity,
to name but two), and are therefore essential: the theory of tensors (including
covariant differentiation of them); Riemannian curvature; geodesies and the
calculus of variations (including the conservation laws and Hamiltonian
formalism); the particular case of skew-symmetric tensors (i.e. "forms")
together with the operations on them; and the various formulae akin to
Stokes' (including the all-embracing and invariant" general Stokes formula "
in n dimensions).